Getting the number of faces a certain vertex is part of
a very nice feature of bmesh :)

import bpy
import bmesh

me = bpy.context.object.data


# Get a BMesh representation
bm = bmesh.new()   # create an empty BMesh
bm.from_mesh(me)   # fill it in from a Mesh

for f in bm.faces:
    for v in f.verts:
        lf = v.link_faces
        print('num faces:', len(lf), 'faces: ', lf[:])
    break